Bending a mesh along spline which it's also following,not working
I don't know if this will work for what you finally want, but for what you described, I would simply use the curve modifier and then animate moving the object along the X axis. Here's what I did


add the cylinder, subdivide it, rotate it, apply rotation and move the origin, as you did.
add a Bezier curve.
in Edit mode, position the Bezier curve to create the path I wanted
add the curve modifier to the cylinder using X as the deformation Axis.


This will move the cylinder to the starting point of the Bezier curve.  Key frame it here.

Move the time to whatever is appropriate, move the cylinder along the x axis, and keyframe.